To solve this problem, we need to find the first five terms of a geometric sequence. We are given the sixth term $a_6 = 25$ and the eighth term $a_8 = 6.25$. The formula for the $n$-th term of a geometric sequence is:

$a_n = a_1 \cdot r^{n-1}$

Where:

• $a_n$ is the $n$-th term.
• $a_1$ is the first term.
• $r$ is the common ratio.

Step 1: Set up two equations for $a_6$ and $a_8$

From the general formula, we have:

$a_6 = a_1 \cdot r^5 = 25$ $a_8 = a_1 \cdot r^7 = 6.25$

Step 2: Solve for the common ratio $r$

To eliminate $a_1$, divide the second equation by the first:

$\frac{a_8}{a_6} = \frac{a_1 \cdot r^7}{a_1 \cdot r^5} = r^2$

This simplifies to:

$\frac{6.25}{25} = r^2$ $r^2 = 0.25$ $r = 0.5$

Step 3: Solve for $a_1$

Now substitute $r = 0.5$ into the equation for $a_6$:

$a_1 \cdot (0.5)^5 = 25$ $a_1 \cdot \frac{1}{32} = 25$ $a_1 = 25 \cdot 32 = 800$

Step 4: Find the first five terms

Now that we know $a_1 = 800$ and $r = 0.5$, we can calculate the first five terms:

$a_1 = 800$ $a_2 = a_1 \cdot r = 800 \cdot 0.5 = 400$ $a_3 = a_2 \cdot r = 400 \cdot 0.5 = 200$ $a_4 = a_3 \cdot r = 200 \cdot 0.5 = 100$ $a_5 = a_4 \cdot r = 100 \cdot 0.5 = 50$

Final Answer:

The first five terms of the geometric sequence are: $a_1 = 800, \quad a_2 = 400, \quad a_3 = 200, \quad a_4 = 100, \quad a_5 = 50$

Let me know if you would like further details or clarifications!

Related Questions:

1. How do you derive the formula for the $n$-th term of a geometric sequence?
2. What happens to the terms of a geometric sequence if the common ratio is greater than 1?
3. How would you solve the problem if you were given different terms of the sequence, say $a_3$ and $a_7$?
4. Can you explain how geometric sequences are used in financial contexts, like compound interest?
5. What is the sum of the first five terms of this geometric sequence?

Tip: In geometric sequences, the ratio between successive terms is always constant, making it easier to detect and calculate unknown terms.